# 2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper

** University Question Papers **

**
**2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper

Attempt any five questions. All questions carry equal marks. Any result

proved in the class may be cited and used without proof.

1. a) Let X be compact and Hausdorff, A ( X be closed. Show that X/A

is homeomorphic to the one-point compactification of X - A.

b) Describe explicitly the quotient topology on the quotient group

IR/|Q, IR being the real line, |Q the set of rationals, treated as a subgroup

of the group (IR, +).

2. a) Prove that GL(n, C) is path connected (hint; use the polynomial

p(z) = det((1 - z)I + zA) for A 2 GL(n, C)).

b) Prove that any discrete subgroup of S1 must necessarily be finite

cyclic.

3. a) Let X be any space. Show that CX, the cone over X is contractible.

b) Show that Sn-1 is a deformation retract of Sn - {N, S},N and S being the north and south poles of Sn respectively.

4. Let f, g : X ! Sn be continuous maps with f(x) 6= -g(x) 8 x 2 X.

Prove that f ' g.

5. Let X be a space. Then show that X is path connected if and only if

all constant maps: X ! X are homotopic to each other.

6. Let R_ : S1 ! S1 be a rotation by angle _. Show that R_ is homotopic

to the identity map: S1 ! S1.

7. Let G be a connected group, H a discrete normal subgroup. Prove that

H _ Z(G), the centre of G.